Inverse design of graded phononic materials via ray tracing

We present a framework for the inverse design of spatially graded phononic materials based on ray tracing. Spatial grading of phononic materials allows the unit cell to vary smoothly in space. Compared to periodic architectures, spatial grading opens up a vast design space that promises new capabilities for manipulating elastic and acoustic waves. However, the use of spatial grading to control wave propagation has been limited to simple gradings, largely due to the bottleneck of modeling efficiency, and to the long-wavelength limit of low frequencies. In this work, ray tracing is exploited as an efficient alternative, around which we develop an optimization framework based on the adjoint state method toward the flexible inverse design of graded phononic materials. We demonstrate the design of graded mass-spring networks for objectives including focusing all directions of a point source, broadband focusing of a plane wave, and frequency sorting. These objectives are out of reach of periodic phononic materials, highlighting the high potential of spatially graded phononic materials. Moreover, our results demonstrate the importance of wave dispersion, which is classically neglected in the long-wavelength limit of elastodynamics.